In a heavy, non-reactive saucepan over medium heat, combine the wine and brown sugar. Cook until reduced to ½ cup, 20 to 25 minutes. Set aside to cool.
In a small bowl mix the butter and rosemary.
Prepare the grill for direct cooking over high heat (450° to 550°F).
In a large bowl gently combine the patty ingredients and shape into four patties of equal size and thickness, each about ¾ inch thick. With your thumb or the back of a spoon, make a shallow indentation about 1 inch wide in the center of each patty.
Brush the cooking grates clean. Grill the patties over direct high heat, with the lid closed as much as possible, until cooked to medium, 8 to 10 minutes, brushing with the glaze every 2 minutes and turning them once when the patties release easily from the grate without sticking. During the last minute of cooking, place a slice of cheese on each patty to melt.
Lower the temperature of the grill to medium heat (350° to 450°F). Brush the tomato slices with oil, season with salt, and grill over direct medium heat until soft, 2 to 4 minutes, turning once. Spread the cut sides of the focaccia with the rosemary butter and grill over direct medium heat, cut sides down, until lightly toasted, about 1 minute. Build each burger with arugula, a patty, a slice of bacon, and 2 slices of tomato. Serve warm.